[Proof of Theorem REF ]
Without loss of generality, we may assume that \(A\)  and \(B\)  are compact sets in \([0, 1]\) . Using Frostman's lemma ([1]}), we can find probability measures \(\mu _A\)  and \(\mu _B\)  supported on \(A\)  and \(B\) , respectively, such that
\(\mu _A(B(x, r))\lesssim r^\alpha , ~\mu _B(B(x,r))\lesssim r^\beta .\) 
